**Jesús Caja García 1,\*, Alfredo Sanz Lobera 2, Piera Maresca 1, Teresa Fernández Pareja <sup>3</sup> and Chen Wang <sup>1</sup>**


Received: 1 July 2018; Accepted: 18 August 2018; Published: 20 August 2018

**Abstract:** Surface metrology employs various measurement techniques, among which there has been an increase of noteworthy research into non-contact optical and contact stylus methods. However, some deeper considerations about their differentiation and compatibility are still lacking and necessary. This work compares the measurement characteristics of the confocal microscope with the portable stylus profilometer instrumentation, from a metrological point of view (measurement precision and accuracy, and complexity of algorithms for data processing) and an operational view (measuring ranges, measurement speed, environmental and operational requirements, and cost). Mathematical models and algorithms for roughness parameters calculation and their associated uncertainties evaluation are developed and validated. The experimental results demonstrate that the stylus profilometer presents the most reliable measurement with the highest measurement speed and the least complex algorithms, while the image confocal method takes advantage of higher vertical and horizontal resolution when compared with the employed stylus profilometer.

**Keywords:** surface texture; contact measurement; optical measurement

### **1. Introduction**

The evaluation of a surface texture involves the analysis of a large number of data using complex models [1]. For this purpose, metrology instruments must scan/measure the surface, obtaining a finite digital sample. Leach et al. [2] establish that the surfaces to be evaluated are defined by the measurement method used (measuring principle). The results will be different even when measuring the same surface with different instruments, since different physical properties are being measured. According to this idea, the instruments used in surface texture measurements can be classified into different groups, depending on the measurement technique and the physical property used to obtain the surface coordinates. When surface texture measurements are carried out on a mechanical manufacturing environment, the two main groups of instruments are based on tactile and optical methods.

In this context, tactile methods are mainly based on the use of stylus profilometers (SP), which are currently the most widely used instruments in the mechanical manufacturing industry [3]. In a common stylus profilometer, a probe, which is in contact with the surface, is physically moved

over it, so that the vertical movement of the tip allows for characterization of the surface heights. This kind of instrument is preferably used in 2-D measurements based on profiles.

On the other hand, optical methods present a wide number of techniques for performing the measurement without surface contact using light instead of a physical probe to measure the surface [4]. The light reflected on the surface and its subsequent detection allows the evaluation of the surface texture. Conroy et al. [5] pointed out that the most widely used optical methods are interferometry and confocal microscopy (CM). The present work is focused on the second one, due to better adequacy when used in mechanical manufacturing environments.

The physical principle of confocal microscopy is based on eliminating the reflected light coming from the out-of-focus planes. The way to achieve this consists of illuminating a small area of the sample. The light beam from the focal plane is then taken so the beams from the lower and upper planes are removed by using a pin-hole. This confocal probe evaluates each point on the surface to be measured and obtains its height and associated light intensity. A system of lateral scanning allows having a line profile and areal measurement. The metrological characteristics of these devices are similar to those provided by the stylus profilometers and are gradually replacing them in specific metrological approaches [6].

On the other hand, different studies have evaluated and compared different techniques of surface metrology, emphasizing the characteristics of the equipment and the differences between the obtained surface parameters but not going into such detail in other non-metrological aspects related with the measurement.

Conroy et al. [5] measure a specimen consisting of an 80 μm pitch square wave Al-coated etched grating with a nominal step height of 187 nm and use stylus profilometers (SP), confocal microscopy (CM), and interferometric microscopy (IM) in their comparison. They do not provide experimental results of the evaluated surface texture parameters nor employed algorithms and conclude that the use of any technique requires an understanding of the properties of the sample, limitations of the technique used, and the analysis required before carrying out the surface measurement. Vorburger et al. [7] compare four techniques including stylus profiling (SP), phase-shifted microscopy (PSIM), white light interferometric microscopy (WLIM), and confocal microscopy (CM). They find discrepancies between WLIM and the other techniques, obtaining similar results among the other three. Poon et al. [8] compare three techniques—stylus profiler (SP), atomic force microscope (AFM), and non-contact optical profiler (NOP)—and conclude with a recommendation on the use of the analyzed techniques when a glass-ceramic substrate is measured. Nouira et al. [9] focused their work on the development of a high-precision profilometer with both optical (CM) and tactile capabilities and measures a VLSI Step-Height Standard (SHS 880-QC). Obtained results in their work show that the tactile measurements, which include stylus profilometers (SP) and atomic force microscopy (AFM), are more accurate than the optical measurements carried out by confocal microscopy (CM). The comparison of both tactile techniques reveals that the SP and the AFM measurements produce very similar results. Piska and Metelkova [10] analyze the relations between 2-D (profile) and 3-D (areal) surface parameters of the same measured surface, and they observed that both methods, SP and CM, give very comparable results only if the surface has a good reflection value. Nielsony et al. [11] analyze differences between a stylus profilometer (SP) and confocal microscopy (CM) in measuring a cladded surface, concluding that CM values of the roughness parameters are higher than SP values. The same result is obtained by Merola et al. [12] analyzing the tribological behavior of retrieved hip femoral head by using a stylus profilometer (SP) and confocal microscopy (CM). These variations in the results can affect the results of the surface topography [13–15], so the measurement principle should be close to the physical functional behavior of the surface [2].

Additional works have analyzed the behavior of optical instruments. Thompson et al. [16] make a quantitative comparison of areal topography measurements by using four optical techniques on a selective laser melting manufactured part. These techniques are confocal microscopy (CM), coherence scanning interferometry (CSI), focus variation microscopy (FVM), and X-ray computed

tomography (XCT). The authors analyze the profile discrepancy between instrument pairs, obtaining high values (near 50%), due to the poorer capture of smaller scale peaks and pits of the FVM instrument. In the same line of work, Feidenhans'l et al. [17] compares optical methods for surface roughness measurement, employing different scatterometers and confocal microscopy (CM). The results are compatible between instruments, but it is necessary to include a Gaussian smoothing function to compensate for the differences.

All these results are interesting, although only partially covering the type of parts that the present work addresses, that is to say, those that are manufactured and used in mechanical manufacturing processes, such as machining processes. Moreover, algorithms or measuring procedures are weakly or not described in all that previous work. For these reasons, this work analyses and compares the application of both techniques, stylus profilometer (SP) and confocal microscopy (CM) measurements, in the evaluation of a series of surfaces, which include machined surfaces and two roughness standards (type C1 and C4 spacing standard), by establishing a comparison between them, and not only considering the results obtained (roughness parameters and their associated uncertainty), but also the procedure and the requirements and performances that these techniques need and offer. The work also considers other aspects related with the measurement, such as the set-up operations of pre-measurement samples, the operating time, operational considerations, data storage requirements, and the cost of instruments and maintenance.

#### **2. Evaluation Procedure of Roughness Parameters**

#### *2.1. Parameters Calculation*

In order to perform the proposed comparison, a specific procedure to evaluate the roughness parameters of a profile from its coordinates (*x*, *z*) has been developed. This way, when the results are compared, only the difference due to the type of instrument, and especially the data acquisition procedure, will be evaluated, which is not affected by systematic effects due to the calculation software that is different for each instrument.

The procedure of obtaining results is defined by the following steps:


$$z = a\mathbf{x} + b \tag{1}$$

The coefficients *a* and *b* can be calculated using the least squares method. The line that best fits the set of coordinates (*xi*, *zi*) is:

$$z\_i - a\mathbf{x}\_i - b + e\_i \approx 0 \tag{2}$$

where *ei* is the residual. It is possible to solve a linear system, obtaining the coefficients *a* and *b*, using:

$$\mathbf{P} = \left(\mathbf{A} \cdot \mathbf{A}^T\right)^{-1} \cdot \mathbf{A}^T \cdot \mathbf{B} \quad \text{where} \quad \mathbf{A} = \begin{bmatrix} x\_1 & 1 \\ \vdots & \vdots \\ x\_n & 1 \end{bmatrix} \quad \mathbf{B} = \begin{bmatrix} z\_1 \\ \vdots \\ z\_n \end{bmatrix} \quad \mathbf{P} = \begin{bmatrix} a \\ b \end{bmatrix} \tag{3}$$

Using the same procedure, it is possible to eliminate the form of surfaces that can be adjusted to quadratic polynomials.


$$R\_p = \max\_{1 \le i \le m} z\_{pi} \tag{4}$$

$$R\_{\upsilon} = \max\_{1 \le i \le m} z\_{\upsilon i} \tag{5}$$

$$R\_z = R\_p + R\_v \quad \Rightarrow \text{ all calculated over a sampling length} \tag{6}$$

*Rt* = *Rp* + *Rv* ⇒ all calculated over the evaluation length (7)

$$R\_a = \frac{1}{l} \int\_0^l |z(\mathbf{x})| d\mathbf{x} \quad \approx \quad R\_a = \frac{1}{n} \sum\_{i=1}^n |z\_i| \tag{8}$$

$$R\_{\emptyset} = \sqrt{\frac{1}{l} \int\_0^l z^2(\mathbf{x})d\mathbf{x}} \quad \approx \quad R\_{\emptyset} = \frac{1}{n} \sum\_{i=1}^n z\_i^2 \tag{9}$$

$$R\_{sk} = \frac{1}{R\_q^3} \left[ \frac{1}{l} \int\_0^l z^3(x) dx \right] \quad \approx \quad R\_{sk} = \frac{1}{R\_q^3} \frac{1}{n} \sum\_{i=1}^n z\_i^3 \tag{10}$$

$$R\_{ku} = \frac{1}{R\_q^4} \left[ \frac{1}{l} \int\_0^l z^4(x) dx \right] \quad \approx \quad R\_{ku} = \frac{1}{R\_q^4} \frac{1}{n} \sum\_{i=1}^n z\_i^4 \tag{11}$$

Parameters *Rt*, *Rz*, *Rp*, and *Rv* measure the amplitude of the profile (peak and valley distances), *Ra* is used to calculate the average roughness, *Rq* measures the variance of the amplitude distribution function (ADF) of the profile, *Rsk* analyze the asymmetry of the ADF, and *Rku* evaluates the spikiness of the profile. Depending on the type of instrument evaluated, different coordinates (*x*, *z*) will be generated, so it will not be necessary to execute all steps.

#### *2.2. Model for Calculating Uncertainties*

Following the calculation model of roughness parameters described in the previous section, for the calculation of the uncertainty of these parameters, the Monte Carlo method is used. This numerical resolution method has a high application in metrological fields [21–23].

The documents Supplement 1 to the "Guide to the expression of uncertainty in measurement (GUM)"—Propagation of distributions using a Monte Carlo method [24] and Supplement 2 to the "Guide to the expression of uncertainty in measurement (GUM)"—Extension to any number of output quantities [25] have been used for the development of the calculation algorithms.

The calculation algorithms are provided in the following steps:


taken into account. All of these magnitudes analyze the variations generated by the instrument in the measurements: noise in the readings, imperfections in the reference of the instrument, sampling and digitizing process, and rounding-off of the coordinates and software calculations, as well as the horizontal and vertical resolution of the instrument and the idealization of the Gaussian filter [26,27].

3. Assignment of the probability density functions (PDF) to the input variables. For the input variables defined above, it is established that variability due to the process of obtaining the sampled coordinates responds to a normal distribution with a mean of the raw coordinate value and a standard deviation equal to 2% of the sampling step [28,29]. In order to analyze the variability of the z-coordinates, it was necessary to perform a repeatability study of the measurements [30]. That is to say, measuring the same profile a large number of times, by the same operator, with the same measurement procedure, same measuring system, and same operation conditions [31]. After doing different experiments, it has been observed that it is practically impossible to obtain the same position of the z-coordinate to be analyzed. Therefore, an analysis of the repeatability of the parameters *Pp* and *Pv* is planned, where the parameters *Pp* and *Pv* are the maximum profile peak height of the primary profile and maximum profile valley depth of the primary profile, respectively. For this type C1 spacing standard, grooves having a sine wave profile will be measured and the standard deviations *S* of the previous parameters will be determined. This study assumes that this variability corresponds to a *t*-distribution (employed when a series of indications are evaluated). The standard uncertainty associated with this variability can be estimated as:

$$u(\mathbf{x}) = \sqrt{\frac{\nu\_p}{\nu\_p - 2}} \frac{S}{\sqrt{n}} \tag{12}$$

where *S* is the maximum standard deviation of the parameters *Pp* and *Pv*, obtained in the experiment, *ν<sup>p</sup>* is the degrees of freedom of the obtained parameters, and *n* represents the numbers of measurements made in the roughness measurement (typically one). Therefore, the digitized coordinates respond to a *t*-distribution of the mean value of the raw coordinate *z* and a standard uncertainty equal to the value calculated by the previous equation. The experimental values obtained are shown in Section 4. The scale division errors of the instrument on the z-axis, responds to a rectangular distribution of limits [−*E*, *E*], where *E* is the scale division on the z-axis.


#### *2.3. Algorithm Validation*

In order to determine the validity and accuracy of the developed algorithms, both reference and synthetic data sets [34] have been used. These data are the coordinates (*x*, *z*) of a series of points linked to a reference profile. The reference data have been obtained from the Internet-based Surface Metrology Algorithm Testing System of the National Institute of Standard and Technology (NIST) [29]. These simulation-generated profiles and F1-type software standards have been generated in accordance with the specifications of ISO 5436-2 [35].

One of the verifications performed in this work is shown in Table 1. The F1-type standard Mill.sdf file [29] shown in Figure 1 has been used and its roughness parameters have been evaluated using a cut-off (*λc*) of 0.8 μm and a Gaussian filter according to ISO 16610-21 [18] (steps 4 and 5).


**Table 1.** Reference values vs calculated values (NIST Mill measured profile).

**Figure 1.** Mill F1-type (**a**) primary and waviness profile; (**b**) roughness profile.

In order to determine how good the developed algorithms are, the calculated results with the reference data are compared with NIST data. The difference, in absolute value, between the parameters obtained by the proposed algorithms and the reference parameters is used.

$$Q\_1 = \left| R^{calculated} - R^{reference} \right| \tag{13}$$

In view of the results, it can be observed that the maximum percentage difference was 0.0061%, which is the reason why we claim the developed algorithms behave in a totally satisfactory manner. In addition, the algorithms of form elimination and low pass filtering for *λ<sup>s</sup>* (steps 2 and 3) using the software provided by the Physikalisch-TechnischeBundesanstalt (PTB) "Software to Analyse Roughness of Profiles-Version 2.09" [36] have been validated, since they provide similar results to those shown. A Type D roughness standard has been measured, and the data of the extracted profile has been introduced in the previous software, obtaining the following results (Table 2).


